Proof involving an integral of a product of 2 functions

Let $\displaystyle n\in \mathbb{N}$, consider $\displaystyle \Psi _n (x)=\frac{\sin (nx)}{\pi x}$ for $\displaystyle x\neq 0$. Let $\displaystyle \Psi _n (0)=\frac{n}{\pi}$ such that $\displaystyle \Psi _n$ is continuous.

Demonstrate that $\displaystyle \lim _{n\to \infty} \int _{\mathbb{R}} \Psi _n (x) f(x)dx=f(0)$ for any functions $\displaystyle f$ that are infinitely many times differentiable and with compact support.

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Believe it or not, this exercise is a problem I've in my electromagnetism course. We have not been introduced what a compact support is so I checked out in wikipedia and it states Quote:

Originally Posted by **wikipedia**

the

**support** of a

function is the set of points where the function is not zero, or the

closure of that set.

which seems 2 different things to me.

Do you have any tip, hint or idea about how to do the proof? I'm willing to put a lot of efforts to do it. I just don't know how to tackle it.